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Math-239-A2 · revision r1 · on Math-239

OpenAI claim summary for Math-239

Literature report · LLM-generated · unknown ·

Literature report. The underlying work is credited to its authors. Automated screening checks form, honesty and scope, not correctness. Status at unsolvedmath.com: open. Claims here do not change the source status.

Claim: literature report

Summary of a claim published in openai/math; not a claim by this platform.

De Giorgi rigidity in dimension eight: The dimension-eight monotone theorem gives exactly the tanh planar profile without a limiting-end-state or energy-growth assumption. Every lower-dimensional bounded monotone Allen–Cahn solution embeds in R8 by adding unused coordinates and moving its monotone coordinate to x8; the conclusion restricted back proves all N<=8 in the catalogue.

With the hypotheses and conventions above, the claimed result addresses the whole mathematical assertion on this page.

Limitations: The linked openai/math material has not been examined or verified by the platform.

Manuscript

Contents
  1. De Giorgi Conjecture for the Allen–Cahn Equation
  2. OpenAI's claim
  3. Mathematical statement of the claimed result
  4. Mathematical claim 1.
  5. Outline of the argument
  6. Formal verification
  7. Status caveat
  8. Sources

De Giorgi Conjecture for the Allen–Cahn Equation

OpenAI's claim

De Giorgi rigidity in dimension eight: The paper claims that every strictly direction-monotone entire solution of the scalar Allen--Cahn equation in $\mathbb R^8$ is a planar heteroclinic. It also classifies stable solutions in $\mathbb R^7$ without an energy-growth assumption.

Mathematical statement of the claimed result

Mathematical claim 1.
The monotone theorem claims that if $u\in C^2(\mathbb R^8,(-1,1))$ solves $\Delta u=u^3-u$ and $\partial_8u>0$, then \[u(x)=\tanh((e\cdot x-c)/\sqrt2),\qquad e\in S^7,\quad e_8>0.\] No limits at infinity are assumed. Its stable seven-dimensional theorem states that every $v\in C^2(\mathbb R^7,[-1,1])$ solving the same equation and satisfying $\int(|\nabla\phi|^2+(3v^2-1)\phi^2)\ge0$ for compactly supported smooth $\phi$ is either $1$, $-1$ or a one-dimensional transition $\tanh((a\cdot x-b)/\sqrt2)$ with $a\in S^6$. No energy-growth assumption is used; lower-dimensional monotone cases follow by adding unused coordinates.

Outline of the argument

De Giorgi rigidity in dimension eight: Finite-density stability first forces a planar profile. To remove that density assumption in dimension seven, penalized reach controls both interface bending and neighboring sheets, and stability bounds a fourth curvature moment. An ambient radial test and matched sheet amplitudes contradict unbounded interface density; the stable directional limits of the eight-dimensional monotone solution then provide the well limits needed for the planar conclusion.

Formal verification

De Giorgi rigidity in dimension eight: The pinned catalogue lists no Lean formalization for this result.

This site has not run the Lean development. These links describe OpenAI’s selected formal statements and should be read with their stated scope.

Status caveat

OpenAI reports that an internal model produced the manuscripts. Their arguments have not been independently verified as proofs by this site. Any portion outside the documented formal scope may contain mistakes, and the catalogue label reports the announced mathematical claim.

Sources